Research in Algebra

Including number theory, algebraic geometry, and combinatorics

We have large groups of researchers active in number theory and algebraic geometry, as well as many individuals who work in other areas of algebra: groups, noncommutative rings, Lie algebras and Lie super-algebras, representation theory, combinatorics, game theory, and coding. A number of members of the algebra group belong to the Research Training Group in Representation Theory, Geometry and Combinatorics, which runs activities and supports grad students and postdocs in its areas of interest.

Courses

Undergraduate upper division courses

Math 110 (and honors version, Math H110). Linear algebra.
Math 113 (and honors version, Math H113). Introduction to abstract algebra.
Math 114. Second course in abstract algebra. (Instructor's choice; usually Galois Theory)
Math 115. Introduction to number theory.
Math 116. Cryptography.
Math 143. Elementary algebraic geometry.
Math 172. Combinatorics.

The Mathematics Department also offers, at the undergraduate level, courses which may include algebraic topics along with others: Problem Solving (H90), Experimental Courses (191), a Special Topics course (195), and several courses of directed and independent individual and group work (196-199).

Graduate courses

Math 245A. General theory of algebraic structures.
Math 249. Algebraic combinatorics.
Math 250A. Groups, rings and fields.
Math 250B. Multilinear algebra and further topics.
Math 251. Ring theory.
Math 252. Representation theory.
Math 253. Homological algebra.
Math 254A-254B. Number theory.
Math 255. Algebraic curves.
Math 256A-256B. Algebraic geometry.
Math 257. Group theory.

Seminars

2008-2009

Math 290. Seminar - Commutative algebra and algebraic geometry, David Eisenbud
Math 290. Seminar - Number theory, Kenneth Ribet

Spring 2009

Math 274. Topics in Algebra - Tropical geometry, Bernd Sturmfels
Math 274. Topics in Algebra - Infinitesimal geometry, Mariusz Wodzicki

Fall 2008

Math 290. Seminar - Algebraic geometry, David Eisenbud and Daniel Erman
Math 290. Seminar - Discrete mathematics, Bernd Sturmfels
Math 290. Seminar - Representation theory, geometry and combinatorics, Mark Haiman and Nicolai Reshetikhin
Math 290. Seminar - Student arithmetic geometry seminar, Martin Olsson
Student Seminar. Student algebraic and arithmetic geometry seminar, David Brown, Daniel Erman and Anthony Varilly

2007-2008

Math 270. Hot Topics - Derived algebraic geometry and topology, Peter Teichner
Math 290. Seminar - Commutative algebra and algebraic geometry, David Eisenbud
Math 290. Seminar - Representation theory, geometry and combinatorics, Mark Haiman and Nicolai Reshetikhin

Spring 2008

Math 274. Topics in Algebra - Real p-adic analysis, Robert Coleman

Fall 2007

Math 274. Topics in Algebra - Locally finite lie algebras and their representations with a view toward open problems, Ivan Penkov
Math 290. Seminar - Student representation theory, geometry and combinatorics, Vera Serganova and Peter Tingley
Math 290. Seminar - Number theory, Kenneth Ribet
Math 290. Seminar - Perverse sheaves, Joel Kamnitzer and Xinwen Zhu

Earlier years, from 2006-2007

 

Senate Faculty

Name Title Research Interests
Matthew Baker Professor Arithmetic Geometry
George M. Bergman Professor Emeritus, Professor of the Graduate School Associative rings, Universal algebra and category theory, Counterexamples
Richard E. Borcherds Professor Lie algebras, Vertex algebras, Automorphic forms
Robert F. Coleman Professor Arithmetic geometry
David Eisenbud Professor Algebraic geometry, Commutative algebra, Computation
Edward Frenkel Professor Representation theory, Integrable systems, Mathematical physics
Mark D. Haiman Professor Algebra, combinatorics, and algebraic geometry
Robin C. Hartshorne Professor Emeritus Algebraic geometry, History of geometry
Tsit-Yuen Lam Professor Emeritus, Professor of the Graduate School Algebra
Hendrik W. Lenstra, Jr. Professor Emeritus Algebraic number theory, Algorithms
Andrew P. Ogg Professor Emeritus Number theory, Elliptic curves, Modular forms
Arthur E. Ogus Professor Algebraic geometry
Martin Olsson Professor Algebraic and arithmetic geometry
Nicolai Reshetikhin Professor Mathematical physics, Low-dimensional topology, Representation theory
John L. Rhodes Professor Emeritus Algebra, Semigroups, Automata
Kenneth A. Ribet Professor Algebraic number theory, Algebraic Geometry
Ichiro Satake Professor Emeritus Symmetric spaces, Automorphic functions
Vera Serganova Professor Super-representation theory
Bernd Sturmfels Professor Combinatorics, Computational algebraic geometry
Constantin Teleman Professor Lie groups, Algebraic geometry, Topology, Quantum field theory
Paul A. Vojta Professor Diophantine approximation, Nevanlinna theory-especially as related to diophantine approximation, Arakelov theory
Lauren Williams Assistant Professor Combinatorics, and connections to representation theory, geometry, and statistical physics
Mariusz Wodzicki Professor Non-commutative and algebraic geometry, Analysis, K-theory

Visiting Faculty

Name Title Research Interests
Mei-Chu Chang Visiting Scholar Combinatorial number theory
Zvezdelina Stankova Visiting Professor & Director, Berkeley Math Circle Algebraic geometry, representation theory, combinatorics, Olympiad problem solving, Berkeley Math Circle

Postdocs

Name Title Research Interests
Valerie Hower Postdoctoral Fellow computational biology, topological and probabilistic methods in data analysis
Nathan Ilten Morrey Assistant Professor Toric geometry and deformation theory
Christopher Manon Postdoctoral Fellow Combinatorial commutative algebra
Alice Medvedev Visiting Assistant Professor (FRG) Model Theory, Algebraic Dynamics
Scott Morrison Miller Research Fellow Topological quantum field theories, Khovanov homology, blob homology; fusion categories, subfactors
Jie Sun Postdoctoral Fellow Infinite Dimensional Lie Theory, Algebraic Geometry, Representation Theory.
Kelli Talaska Postdoctoral Fellow Centered in algebraic combinatorics, with a special focus on connections between matrix algebra and enumeration in planar networks.

Faculty with Related Research Interests

Name Title Research Interests
Denis Auroux Professor Symplectic topology and mirror symmetry
Elwyn R. Berlekamp Professor Emeritus Combinatorial game theory, Algebraic coding theory, Electrical engineering, Computer science
Ralph McKenzie Professor Emeritus General algebra, Logic
Calvin C. Moore Professor Emeritus Representations and actions of topological groups, Operator algebras
Lior Pachter Professor Mathematical and computational biology
Marc A. Rieffel Professor Non-commutative harmonic analysis, Operator algebras, Quantum geometry
Thomas Scanlon Professor Model theory and applications to number theory

Graduate Students

Name Dissertation Supervisor
Piotr Achinger
Natth Bejraburnin Lior Pachter
David Berlekamp David Eisenbud
Shuchao Bi Olga Holtz
Adam L. Boocher David Eisenbud
Morgan Brown David Eisenbud
Melody Chan Bernd Sturmfels
Anastasia Chavez
Harrison Chen
Charles Crissman David Eisenbud
Andrew J. Critch Bernd Sturmfels
Michael W. Daub Kenneth A. Ribet
Jeffrey Doker Federico Ardila at SFSU, Matthias Beck at SFSU, Lior Pachter, Bernd Sturmfels, Lauren Williams
Daniel Halpern-Leistner Constantin Teleman
Katrina Honigs Martin Olsson
Yuhao Huang Martin Olsson
Sarah Iveson Mark D. Haiman
Theodore Johnson-Freyd Nicolai Reshetikhin
Aaron Kleinman Lior Pachter
Alex Kruckman Thomas Scanlon
Chul Hee Lee Richard E. Borcherds
Lasse Peter Mannisto Martin Olsson
Lilit Martirosyan Vera Serganova
James McIvor
George W. Melvin Constantin Teleman
Khoa Nguyen Kenneth A. Ribet
Andrew Niles Martin Olsson
Jae-Young Park Martin Olsson
Eric C. Peterson Constantin Teleman
Edgard Rincon Olga Holtz, Bernd Sturmfels
Douglas P. Rizzolo James Pitman
William Slofstra Constantin Teleman
Pablo Solis Constantin Teleman
Matthew Tucker-Simmons Marc A. Rieffel, Nicolai Reshetikhin
Thanh Quang Vu
Harold M. Williams Nicolai Reshetikhin
Hwajong Yoo Kenneth A. Ribet

Recent Ph.D.s

Name Dissertation Title Dissertation Supervisor Year
Anton Igorevich Geraschenko Toric Stacks Vera Serganova 2011
An Huang On Conformal Field Theory and Number Theory Richard Borcherds 2011
Claudiu Cristian Raicu Secant Varieties of Segre-Veronese Varieties David Eisenbud 2011
Cynthia Leslie Vinzant Real Algebraic Geometry in Convex Optimization Bernd Sturmfels 2011
David Michael Brown Rigid Colomology of Algebraic Stacks Kenneth A Ribet 2010
Dustin Alexander Cartwright Applications of non-linear algebra to biology Bernd Sturmfels 2010
Li-Chung Chen Skew-Linked Partitions and a Representation-Theoretic Model for K-Schur Mark Haiman 2010
Maria Angelica Cueto Tropical implicitization Bernd Sturmfels 2010
Daniel Erman Applications and Extensions of Boij-Soderberg Theory David Eisenbud 2010
Alexander Fink Matroid Polytope Subdivisions and Valuations Bernd Sturmfels 2010
Manuel Lionel Reyes One-Sided Prime Ideals in Noncommutative Algebra Tsit-Yuen Lam 2010
Matthew Satriano Stacky Resolutions of Singular Schemes Martin Olsson 2010
Shenghao Sun On I-Adic Cohomology of Artin Stacks: L-Functions, Weights and the Decomposition Theorem Martin Olsson 2010
Bianca Viray The Algebraic Brauer-Marnin Obstruction on Chatelet Surfaces, Degree 4 del Pezzo Surfaces, and Enriques Surfaces Bjorn Poonen 2010
Jonah Blasiak Cyclage, catabolism, and the affine Hecke algebra Mark Haiman 2009
Ishai Dan-Cohen Moduli of nondegenerate unipotent representations Martin Olsson, Arthur Ogus 2009
Seth Dutter Logarithmic Jet Spaces and Intersection Multiplicities Paul Vojta 2009
Sevak Mkrtchyan Scaling limits of random skew plane partitions Nicolai Reshetikhin 2009
Brian Rothbach Borerl orbits of x2 = ) in gln Mark Haiman 2009
Arun Sharma The Structure of 3-free Permutations Mark Haiman, Ron Graham 2009
Anthony Varilly-Alvarado Arithmetic of del Pezzo surfaces of degree 1 Bjorn Poonen 2009
Xiwen Zhu Gerbal representations of double loop groups Edward Frenkel 2009
Jameel Al-Aidroos Perfect Pairings in the Tautologicl Rings of the Moduli Spcaes of Stable Curves Graber, David Eisenbud 2008
Jennifer Danae Berg On the center of the Lie superalgebra q(n)(2) Vera Serganova 2008
Elizabeth Dan-Cohen Structure of root-reductive Lie algebras Vera Serganova, Joseph A. Wolf 2008
Jesse Fern Calculations of Quantum Error Correction and Fault Tolerance Thresholds Vera Serganova, Birgitta Whaley 2008
David Stephen Freeman Constructing Abelian Varieties for Pairing-Based Cryptography Kenneth Ribit, Edward Schaeffer 2008
Peter McKinley Huggins Polytopes in Computational Biology Lior Pachter, Bernd Sturmfels 2008
Dimitar Petkov Jetchev CM Points, Selmer groups, component groups and Euler Systems Kenneth Ribit 2008
Peter William Tingley Some results on the crystal commutor and affine sl(n) crystals Nicolai Reshetikhin 2008
David James Zywina The large sieve and Galois representations Bjorn Poonen 2008
Sami Hayes Assaf Dual Equivalence Graphs, Ribbon Tableaux and Macdonald Polynomials Mark Haiman 2007
Ioan Berbec Group Schemes Over Artinian Rings and Applications Robert Coleman 2007
Scott Huai-Lei Carnahan Monstrous Lie Algebras and Generalized Moonshine Richard E. Borcherds 2007
Aaron Paul Gray Functoriality of the Logarithmic Riemann-Hilbert Arthur E. Ogus 2007
Aaron Greicius Ellicptic Curves with Surjective Global Galois Representation Bjorn Poonen 2007
Crystal Faye Hoyt Kac-Moody Superalgebras of Finite Growth Vera Serganova 2007
Grace Lyo Semilinear Actions of Galois Groups and the Algebraic K-Theory of Fields Gunnar Carlsson 2007
Carl Alexander Miller Cohomology of p-torsion sheaves on characteristic-p curvers Arthur E. Ogus 2007
Jason Ryder Morton Geometry of Conditional Independence Bernd Sturmfels 2007
Benjamin Thomas Webster Poisson Algebraic Geometry in Representation Theory and Combinatorics Nicolai Y. Reshetikhin 2007
Jared Seth Weinstein Automorphic Representations with Local Constraints Kenneth A. Ribet 2007
Soroosh Yazdani Modular Abelian Varity of Odd Modular Degree Kenneth A. Ribet 2007
Josephine Thi Mar Lwin Yu Combinatorial Aspects of Tropical Geometry Bernd Sturmfels 2007